In Pierre there is VHA (Varmint Hunters Assoc.). They have a web site and 800 number and if you want an experience and don't mind a bit of cost... Membershjp. Magazine. Parties/shooting matches if you want. Look them up.
There has been a drought. Populations are, no doubt, down a bit.
South Center SD is the Rosebud Rez (reservation). Rosebud Sioux Indians. They prefer name Lakota, but that is another story. They have a "game department" that controls hunting/shooting on the Rez. Google them and get the most current rules. Same story, North Center of SD. Cheyenne River Sioux. Eagle Butte. Many are very pleased with these folks. Lakotamall.com is run from the Pine Ridge Rez more to the SW corner of SD. They have a drinking problem [Rez is dry and Whiteclay NE just across the border sells millions of cans of beer a year. To whom you might ask? Cowboys? Ha, ha, ha] and some unhappy stories have come from there. People not keeping their commitments. But "dog country." South of Wall there is Badlands nat. Park. [NO SHOOTING IN THE PARK. Fed slam.] South of park is Buffalo Gap Nat. Grasslands. Public land free to all.
But. There has been a ton of crap about p'dogs being endangered. (not true but part of a tree hugger hustle). Areas have been closed. Shoot there, fed slam. And they want to re establish the black footed ferret, a weasel that was almost wiped out by a house cat disease... If you are going, on your own, find these areas and avoid them... Or suffer... forest Service has had a "kiosk" just off the interstate (70? I never remember if it is 70 or 90... Hair is thinning too...) at Wall in the summer.
And there was a list of ranchers beggin for shooters put out by extension in SD. Haven't seen a copy for years, but you might ask around. luck [Wyoming is better IMHO, but suit yourself.]